Aspects of N=4 SYM

TL;DR

This work develops a comprehensive, non-perturbative treatment of gauge-invariant operators and their correlation functions in N=4 SYM using analytic superspace. It provides a detailed operator classification into protected, semi-protected, and long representations, clarifies how shortening conditions and reducibility at threshold govern renormalisation, and introduces the E tensor to construct superconformal invariants and track U(1)_Y properties. By solving PSL(4|4) Ward identities for two-, three-, and higher-point functions, it establishes non-renormalisation results for protected operators and exposes subtle violations of U(1)_Y in semi-protected or long cases. The paper also presents coordinate and Grassmannian methods to generate all superconformal invariants and discusses the extension to harmonic superspace, highlighting the practical impact for understanding non-perturbative structure and potential AdS/CFT implications.

Abstract

The properties of gauge-invariant composite operators and their correlation functions in N=4 SYM are discussed in the analytic superspace formalism. A complete classification of the different types of operators in the theory is given. Operators can be either protected or unprotected according to whether they do not or do have anomalous dimensions, and the analytic superspace formalism allows one to identify which type a given operator is in a straightforward manner. A simple discussion is given of the behaviour of reducible multiplets at threshold. It is pointed out that there is a class of ``semi-protected'' operators which do not have anomalous dimensions but which do not necessarily have non-renormalised three-point functions when the other two operators in the correlator are protected, although two-point functions of such operators are non-renormalised. A complete discussion of superconformal invariants in analytic superspace is given. The paper includes a modified discussion of the transformation rules of analytic superfields which clarifies the $U(1)_Y$ properties of operators and correlation functions and, in particular, explicit examples are given of three-point correlation functions which violate this symmetry. A tensor, $\cE$, invariant under $SL(n|m)$ but not under $GL(n|m)$, is introduced and used in the discussion of $U(1)_Y$ and in the construction of invariants.